Solve Using the Quadratic Formula (n+7)^2=-2n

${(n + 7)}^{2} = - 2 n$

Step 1

Move all terms to the left side of the equation and simplify.

Add $2 n$ to both sides of the equation.

${(n + 7)}^{2} + 2 n = 0$

Simplify ${(n + 7)}^{2} + 2 n$ .

Simplify each term.

Rewrite ${(n + 7)}^{2}$ as $(n + 7) (n + 7)$ .

$(n + 7) (n + 7) + 2 n = 0$

Expand $(n + 7) (n + 7)$ using the FOIL Method.

Apply the distributive property.

$n (n + 7) + 7 (n + 7) + 2 n = 0$

Apply the distributive property.

$n \cdot n + n \cdot 7 + 7 (n + 7) + 2 n = 0$

Apply the distributive property.

$n \cdot n + n \cdot 7 + 7 n + 7 \cdot 7 + 2 n = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $n$ by $n$ .

$n^{2} + n \cdot 7 + 7 n + 7 \cdot 7 + 2 n = 0$

Move $7$ to the left of $n$ .

$n^{2} + 7 \cdot n + 7 n + 7 \cdot 7 + 2 n = 0$

Multiply $7$ by $7$ .

$n^{2} + 7 n + 7 n + 49 + 2 n = 0$

Add $7 n$ and $7 n$ .

$n^{2} + 14 n + 49 + 2 n = 0$

Add $14 n$ and $2 n$ .

$n^{2} + 16 n + 49 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = 16$ , and $c = 49$ into the quadratic formula and solve for $n$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (1 \cdot 49)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $16$ to the power of $2$ .

$n = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$n = \frac{- 16 \pm \sqrt{256 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$n = \frac{- 16 \pm \sqrt{256 - 196}}{2 \cdot 1}$

Subtract $196$ from $256$ .

$n = \frac{- 16 \pm \sqrt{60}}{2 \cdot 1}$

Rewrite $60$ as $2^{2} \cdot 15$ .

Factor $4$ out of $60$ .

$n = \frac{- 16 \pm \sqrt{4 (15)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$n = \frac{- 16 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{- 16 \pm 2 \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{- 16 \pm 2 \sqrt{15}}{2}$

Simplify $\frac{- 16 \pm 2 \sqrt{15}}{2}$ .

$n = - 8 \pm \sqrt{15}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = - 8 \pm \sqrt{15}$

Decimal Form:

$n = - 4.12701665 \dots, - 11.87298334 \dots$

Solve Using the Quadratic Formula (n+7)^2=-2n

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